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We study the memory requirements of self-stabilizing leader election (SSLE) protocols. We are mainly interested in two types of systems: anonymous systems and id-based systems. We consider two classes of protocols: deterministic ones and randomized ones.We prove that a non-constant lower bound on the memory space is required by a SSLE protocol on unidirectional, anonymous rings (even if the protocol is randomized).We show that, if there is a deterministic protocol solving a problem on id-based systems where the processor memory space is constant and the id-values are not bounded then there is a deterministic protocol on anonymous systems using constant memory space that solves the same problem. Thus impossibility results on anonymous rings (i.e. one may design a deterministic SSLE protocol, only on prime size rings, under a centralized daemon) can be extended to those kinds of id-based rings. Nevertheless, it is possible to design a silent and deterministic SSLE protocol requiring constant memory space on unidirectional, id-based rings where the id-values are bounded. We present such a protocol.We also present a randomized SSLE protocol and a token circulation protocol under an unfair, distributed daemon on anonymous and unidirectional rings of any size. We give a lower bound on memory space requirement proving that these protocols are space optimal. The memory space required is constant on average.

Population protocols are a popular model of distributed computing , in which n agents with limited local state interact randomly, and cooperate to collectively compute global predicates. Inspired by recent developments in DNA programming, an extensive series of papers, across different communities, has examined the com-putability and complexity characteristics of this model. Majority, or consensus, is a central task in this model, in which agents need to collectively reach a decision as to which one of two states A or B had a higher initial count. Two metrics are important: the time that a protocol requires to stabilize to an output decision, and the state space size that each agent requires to do so. It is known that majority requires Ω(log log n) states per agent to allow for fast (poly-logarithmic time) stabilization, and that O(log 2 n) states are sufficient. Thus, there is an exponential gap between the space upper and lower bounds for this problem. This paper addresses this question. On the negative side, we provide a new lower bound of Ω(log n) states for any protocol which stabilizes in O(n 1−c) expected time, for any constant c > 0. This result is conditional on monotonicity and output assumptions, satisfied by all known protocols. Technically, it represents a departure from previous lower bounds, in that it does not rely on the existence of dense configurations. Instead, we introduce a new generalized surgery technique to prove the existence of incorrect executions for any algorithm which would contradict the lower bound. Subsequently, our lower bound also applies to general initial configurations, including ones with a leader. On the positive side, we give a new algorithm for majority which uses O(log n) states, and stabilizes in O(log 2 n) expected time. Central to the algorithm is a new leaderless phase clock technique , which allows agents to synchronize in phases of Θ(n log n) consecutive interactions using O(log n) states per agent, exploiting a new connection between population protocols and power-of-two-choices load balancing mechanisms. We also employ our phase * Dan

International audienceWe present the first (practically) self-stabilizing replicated state machine for asynchronous message passing systems. The scheme ensures that starting from an arbitrary configurations, the replicated state-machine eventually exhibits the desired behaviour for a long enough execution regarding all practical considerations

Abstract. This paper considers the fundamental problem of self-stabilizing leader election (SSLE) in the model of population protocols. In this model, an unknown number of asynchronous, anonymous and nite state mobile agents interact in pairs over a given communication graph. SSLE has been shown to be impossible in the original model. This impossibility can been circumvented by a modular technique augmenting the system with an oracle -an external module abstracting the added assumption about the system. Fischer and Jiang have proposed solutions to SSLE, for complete communication graphs and rings, using an oracle Ω?, called the eventual leader detector. In this work, we present a solution for arbitrary graphs, using a composition of two copies of Ω?. We also prove that the di culty comes from the requirement of self-stabilization, by giving a solution without oracle for arbitrary graphs, when an uniform initialization is allowed. Finally, we prove that there is no self-stabilizing implementation of Ω? using SSLE, in a sense we de ne precisely.

We present a randomized self-stabilizing leader election protocol and a randomized self-stabilizing token circulation protocol under an arbitrary scheduler on anonymous and unidirectional rings of any size. These protocols are space optimal. We also give a formal and complete proof of these protocols. To this end, we develop a complete model for probabilistic self-stabilizing distributed systems which clearly separates the non deterministic behavior of the scheduler from the randomized behavior of the protocol. This framework includes all the necessary tools for proving the selfstabilization of a randomized distributed system: definition of a probabilistic space and definition of the self-stabilization of a randomized protocol. We also propose a new technique of scheduler management through a self-stabilizing protocol composition (cross-over composition). Roughly speaking, we force all computations to have a fairness property under any scheduler, even under an unfair one.

International audiencePopulation protocols are a model presented recently for networks with a very large, possibly unknown number of mobile agents having small memory. This model has certain advantages over alternative models (such as DTN) for such networks. However, it was shown that the computational power of this model is limited to semi-linear predicates only. Hence, various extensions were suggested. We present a model that enhances the original model of population protocols by introducing a (weak) notion of speed of the agents. This enhancement allows us to design fast converging protocols with only weak requirements (for example, suppose that there are different types of agents, say agents attached to sick animals and to healthy animals, two meeting agents just need to be able to estimate which of them is faster, e.g., using their types, but not to actually know the speeds of their types). Then, using the new model, we study the gathering problem, in which there is an unknown number of anonymous agents that have values they should deliver to a base station (without replications). We develop efficient protocols step by step searching for an optimal solution and adapting to the size of the available memory. The protocols are simple, though their analysis is somewhat involved. We also present a more involved result - a lower bound on the length of the worst execution for any protocol. Our proofs introduce several techniques that may prove useful also in future studies of time in population protocols

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